A double-dimensional approach to formal category theory

Whereas formal category theory is classically considered within a $2$-category, in this paper a double-dimensional approach is taken. More precisely we develop such theory within the setting of augmented virtual double categories, a notion extending that of virtual double category by adding cells with nullary target. [...] After this the notion of `weak' Kan extension within an augmented virtual double category is considered, together with three strengthenings. [...] The notion of yoneda embedding is then considered in an augmented virtual double category, and compared to that of a good yoneda structure on a $2$-category; the latter in the sense of Street-Walters and Weber. Conditions are given ensuring that a yoneda embedding $y \colon A \to \hat A$ defines $\hat A$ as the free small cocompletion of $A$, in a suitable sense. In the second half we consider formal category theory in the presence of algebraic structures. In detail: to a monad $T$ on an augmented virtual double category $\mathcal K$ several augmented virtual double categories $T\text-\mathsf{Alg}_{(v, w)}$ of $T$-algebras are associated, [...]. This is followed by the study of the creation of, amongst others, left Kan extensions by the forgetful functors $T\text-\mathsf{Alg}_{(v, w)} \to \mathcal K$. The main motivation of this paper is the description of conditions ensuring that yoneda embeddings in $\mathcal K$ lift along these forgetful functors, as well as ensuring that such lifted algebraic yoneda embeddings again define free small cocompletions, now in $T\text-\mathsf{Alg}_{(v, w)}$. As a first example we apply the previous to monoidal structures on categories, hence recovering Day convolution of presheaves and Im-Kelly's result on free monoidal cocompletion, as well as obtaining a "monoidal Yoneda lemma".Comment: Draft. A streamlined and expanded version of Sections 1, 2 and 3 is available as arXiv:1910.11189. v2: main notion 'hypervirtual double category' has been renamed as 'augmented virtual double category'; several new results; some corrections. Comments are welcom

On pointwise Kan extensions in double categories

In this paper we consider a notion of pointwise Kan extension in double categories that naturally generalises Dubuc's notion of pointwise Kan extension along enriched functors. We show that, when considered in equipments that admit opcartesian tabulations, it generalises Street's notion of pointwise Kan extension in 2-categories.Comment: 38 pages. In v2 the proofs of Propositions 4.2 and 4.4 have been simplified; this is the final version, as it appears in Theory and Applications of Categories, Vol. 29, 201

A categorical approach to the maximum theorem

Berge's maximum theorem gives conditions ensuring the continuity of an optimised function as a parameter changes. In this paper we state and prove the maximum theorem in terms of the theory of monoidal topology and the theory of double categories. This approach allows us to generalise (the main assertion of) the maximum theorem, which is classically stated for topological spaces, to pseudotopological spaces and pretopological spaces, as well as to closure spaces, approach spaces and probabilistic approach spaces, amongst others. As a part of this we prove a generalisation of the extreme value theorem.Comment: 45 pages. Minor changes in v2: this is the final preprint for publication in JPA

A Compliment's Cost:How Positive Responses to Non-traditional Choices may Paradoxically Reinforce Traditional Gender Norms

In times of societal change, like changes in gender roles, one may compliment men deciding to spend more time on childcare, or women pursuing a job higher up, to support their pioneering behaviour. However, we predict that while compliments may communicate appreciation of someone's behaviour, they simultaneously communicate that a norm has been breached, and thus that the behaviour is not considered ‘normal’. In four studies (total N = 821), we show that men receive more compliments for reducing work hours for childcare than women (Study 1). Moreover, compliments (compared to neutral responses) signal more descriptive norm deviance, and this has downstream consequences for perceptions of the target's gender belonging and decision doubt and for perceived societal norms (Studies 2–4). Together, these findings contribute to our understanding of normative communication patterns as well as potential paradoxical reinforcement of gender norms through compliments

The group dynamics sparking social change:how group value in diversity predicts interactions with a deviant

Deviants are pivotal to sparking social change but their influence is often hindered by group dynamics that serve to maintain the status quo. This paper examines the influence of a group's value in diversity in deviant's ability to spark social change, with a unique focus on the experience and anticipation of group dynamics that enable minority influence. Hypotheses were tested in three studies (NTotal = 674), which varied in their use of ad-hoc conversation groups or existing friend groups, and whether deviants were newcomers, or existing group members. We demonstrated social influence of a vegan deviant increased to the extent that participants perceived their group to value diversity. Furthermore, group value in diversity related to experienced and anticipated group dynamics that enabled minority influence: decreased conformity pressure, increased attentive listening, and, importantly, an increased search for agreement with the deviant. We discuss the importance of studying group dynamics for understanding what valuing diversity entails